Download Ratio and Proportion

Ratio and Proportion
7-1
EXAMPLE 1
Simplify ratios
Simplify the ratio.
a. 64 m : 6 m
b.
5 ft
20 in.
SOLUTION
a.
Write 64 m : 6 m as 64 m .
6m
Then divide out the units and simplify.
64 m = 32 = 32 : 3
6m
3
b. To simplify a ratio with unlike units, multiply by a
conversion factor.
5 ft = 5 ft
12 in. = 60 = 3
20
1
20 in.
20 in.
1 ft
GUIDED PRACTICE
for Example 1
Simplify the ratio.
1. 24 yards to 3 yards
SOLUTION
24
3
Then divide out the units and simplify.
24
= 8 = 8:1
3
1
Write 24 yards : 3 yards as
GUIDED PRACTICE
for Example 1
Simplify the ratio.
2. 150 cm : 6 m
SOLUTION
To simplify a ratio with unlike units, multiply by a
conversion factors.
150cm = 150
6m
6
1m =
100cm
1
4
= 1:4
EXAMPLE 4
Solve proportions
Solve the proportion.
ALGEBRA
a. 5 = x
10
16
SOLUTION
a.
5
10
x
16
Write original proportion.
5 16 =
10 x
Cross Products Property
80
10 x
Multiply.
x
Divide each side by 10.
=
=
8 =
EXAMPLE 4
b.
Solve proportions
2
1
=
y+1
3y
SOLUTION
b.
1
y+1 =
2
3y
Write original proportion.
1 3y
=
2 (y + 1)
Cross Products Property
3y
=
2y + 2
Distributive Property
y
=
2
Subtract 2y from each side.
GUIDED PRACTICE
5.
for Example 4
2 = 5
x
8
SOLUTION
2
2
x
=
5
8
Write original proportion.
8
=
5 x
Cross Products Property
16 =
5x
Multiply.
x =
16
5
Divide each side by 5 .
GUIDED PRACTICE
6.
for Example 4
1
4
=
x–3
3x
SOLUTION
1
x–3
= 4
3x
Write original proportion.
3x = 4(x – 3)
Cross Products Property
3x = 4x – 12
Multiply.
3x – 4x = – 12
– x = – 12
x = 12
Subtract 4x from each side.
GUIDED PRACTICE
7.
for Example 4
y–3
y
7 = 14
SOLUTION
y–3
y
7 = 14
Write original proportion.
14(y – 3) = 7 y
Cross Products Property
14y – 42 = 7y
Multiply.
14y – 7y = 42
Subtract 7y from each side and
add 42 to each side.
y = 6
Subtract , then divide
EXAMPLE 2
Use a ratio to find a dimension
Painting
You are planning to paint a mural
on a rectangular wall. You know
that the perimeter of the wall is 484
feet and that the ratio of its length
to its width is 9 : 2. Find the area of
the wall.
SOLUTION
STEP 1
Write expressions for the length and width.
Because the ratio of length to width is 9 : 2,
you can represent the length by 9x and the
width by 2x.
EXAMPLE 2
STEP 2
Use a ratio to find a dimension
Solve an equation to find x.
2l + 2w
2(9x) + 2(2x)
22x
x
STEP 3
=
=
=
=
P
484
484
22
Formula for perimeter of rectangle
Substitute for l, w, and P.
Multiply and combine like terms.
Divide each side by 22.
Evaluate the expressions for the length and
width. Substitute the value of x into each
expression.
Length = 9x = 9(22) = 198
Width = 2x = 2(22) = 44
The wall is 198 feet long and 44 feet wide, so its area is
198 ft 44 ft = 8712 ft 2.
EXAMPLE 3
Use extended ratios
ALGEBRA The measures of the angles in
CDE are
in the extended ratio of 1 : 2 : 3. Find the measures of
the angles.
SOLUTION
Begin by sketching the triangle. Then use the
extended ratio of 1 : 2 : 3 to label the measures
as x° , 2x° , and 3x° .
o
o
o
o
Triangle Sum Theorem
x + 2x + 3x = 180
6x = 180
Combine like terms.
Divide each side by 6.
x = 30
ANSWER
o
o
o
o
o
The angle measures are 30 , 2(30 ) = 60 , and 3(30 ) = 90.
GUIDED PRACTICE
for Examples 2 and 3
3. The perimeter of a room is 48 feet and the ratio of
its length to its width is 7 : 5. Find the length and width
of the room.
SOLUTION
STEP 1
Write expressions for the length and width.
Because the ratio of length is 7 : 5, you can
represent the length by 7x and the width by
5x.
for Examples 2 and 3
GUIDED PRACTICE
STEP 2
Solve an equation to find x.
2l + 2w
2(7x) + 2(5x)
24x
x
STEP 3
=
=
=
=
P Formula for perimeter of rectangle
48 Substitute for l, w, and P.
48 Multiply and combine like terms.
2
Evaluate the expressions for the length and
width. Substitute the value of x into each
expression.
Length = 7x + 7(2) = 14 ft
Width = 5x + 5(2) = 10 ft
for Examples 2 and 3
GUIDED PRACTICE
4. A triangle’s angle measures are in the extended
ratio of 1 : 3 : 5. Find the measures of the angles.
SOLUTION
x
Begin by sketching the triangle. Then use the
extended ratio of 1 : 3 : 5 to label the measures
as x° , 2x° , and 3x° .
o
o
o
o
3x
5x
Triangle Sum Theorem
x + 3x + 5x = 180
Combine like terms.
9x = 180
Divide each side by 9.
x = 20
ANSWER
o
o
o
o
o
The angle measures are 20 , 3(20 ) = 60 , and 5(20 ) = 100.